12 research outputs found
Reflecting diffusions and hyperbolic Brownian motions in multidimensional spheres
Diffusion processes moving inside
spheres and reflecting orthogonally on their
surfaces are considered. The stochastic differential equations
governing the reflecting diffusions are presented and their kernels and
distributions explicitly derived. Reflection is obtained by means of the
inversion with respect to the sphere . The particular cases of
Ornstein-Uhlenbeck process and Brownian motion are examined in detail.
The hyperbolic Brownian motion on the Poincar\`e half-space is
examined in the last part of the paper and its reflecting counterpart within
hyperbolic spheres is studied. Finally a section is devoted to reflecting
hyperbolic Brownian motion in the Poincar\`e disc within spheres concentric
with
Reflecting diffusions and hyperbolic Brownian motions in multidimensional spheres
We consider diffusion processes moving inside spheres aS, a"e (d) and reflecting orthogonally on their surfaces. We present stochastic differential equations governing the reflecting diffusions and explicitly derive their kernels and distributions. Reflection is obtained by means of the inversion with respect to the sphere . The particular cases of Ornstein-Uhlenbeck process and Brownian motion are examined in detail. The hyperbolic Brownian motion on the Poincar, half-space a"i (d) is examined in the last part of the paper, and its reflecting counterpart within hyperbolic spheres is studied. Finally, a section is devoted to reflecting hyperbolic Brownian motion in the Poincar, disc D within spheres concentric with D